Question: Is ${245820}$ divisible by $9$ ?
Answer: A number is divisible by $9$ if the sum of its digits is divisible by $9$ . [ Why? First, we can break the number up by place value: $ \begin{eqnarray} {245820}= &&{2}\cdot100000+ \\&&{4}\cdot10000+ \\&&{5}\cdot1000+ \\&&{8}\cdot100+ \\&&{2}\cdot10+ \\&&{0}\cdot1 \end{eqnarray} $ Next, we can rewrite each of the place values as $1$ plus a bunch of $9$ s: $ \begin{eqnarray} {245820}= &&{2}(99999+1)+ \\&&{4}(9999+1)+ \\&&{5}(999+1)+ \\&&{8}(99+1)+ \\&&{2}(9+1)+ \\&&{0} \end{eqnarray} $ Now if we distribute and rearrange, we get this: $ \begin{eqnarray} {245820}= &&\gray{2\cdot99999}+ \\&&\gray{4\cdot9999}+ \\&&\gray{5\cdot999}+ \\&&\gray{8\cdot99}+ \\&&\gray{2\cdot9}+ \\&& {2}+{4}+{5}+{8}+{2}+{0} \end{eqnarray} $ Any number consisting only of $9$ s is a multiple of $9$ , so the first five terms must all be multiples of $9$ That means that to figure out whether the original number is divisible by $9 $ , all we need to do is add up the digits and see if the sum is divisible by $9$ . In other words, ${245820}$ is divisible by $9$ if ${ 2}+{4}+{5}+{8}+{2}+{0}$ is divisible by $9$ Add the digits of ${245820}$ $ {2}+{4}+{5}+{8}+{2}+{0} = {21} $ If ${21}$ is divisible by $9$ , then ${245820}$ must also be divisible by $9$ ${21}$ is not divisible by $9$, therefore ${245820}$ must not be divisible by $9$.